APPENDIX TO CHAPTER X
§ 1 (to Ch. X, § 2)
[Geometric representation of incomes for three years]
IF we proceed from the consideration of two years to that of three, we may still represent our problem geometrically by using a model in three dimensions. Let us imagine three mutually perpendicular axes from an origin O called respectively OX', OX'', OX''', and represent the income combination or income stream for the particular individual by the point P, whose coördinates c', c'', and c''' are the three years' income installments with which the individual is initially endowed. Then through the point P draw, instead of the straight line in the previous representation, a plane ABC cutting the three axes in A, B, and C. This plane has a slope with reference to the two axes OX' and OX'' of equal to 1 + i' (unity or 100 per cent plus the rate of interest connecting the first and second years), and has a slope with reference to the axes OX'' and OX''' represented by equal to 1 + i'' (unity plus the rate of interest connecting the second and third years). Now suppose the space between the axes to be filled with willingness surfaces laminated like the coats of an onion, such that for all points on the same surface, the total desirability or wantability of the triple income combination or income position represented by each of those points will be the same. These surfaces will be such as to approach the three axes and the planes between them, and also such that the attached numbers representing their respective total wantabilities shall increase as they recede from the origin. The plane ABC drawn through P at the slope fixed by the rates of interest just indicated will now be tangent to some one of the willingness surfaces at a point Q, which is the point at which the individual will, under these conditions, fix his income situation, for every point on the plane ABC will have the same present value, and every point on this plane is available to him by borrowing and lending (or buying and selling) at the rates i' and i'', but not all of them will have the same desirability, or wantability. He will select that one which has the maximum wantability, and this will evidently be the point Q, at which the plane is tangent to one of the family of willingness surfaces. This point will be such that the rates of time preference will be equal to the rate of interest.
So much for the individual. The market problem determining the rate of interest is here solved by finding such an orientation for the various planes through the given points called P's as will bring the center of gravity of the tangential points, the Q's, into coincidence with the fixed center of gravity of the P's.
To proceed beyond three years would take us into the fourth dimension and beyond. Such a representation cannot be fully visualized, and therefore has little meaning except to mathematicians.